GATE 2021 AE – Question 60
A two-cell wing box has rectangular cells 200×300 and 300×300 mm, wall thickness 1.5 mm and shear modulus 27 GPa. Under torque 12 kN m, the magnitude of shear stress in common wall AD is ___ MPa (one decimal).
Practise this question in The GATE Grind →
Show answer and explanation
Correct answer: 2.29 to 2.31
Explanation
**Shear flows in a two-cell box.** Let $q_1$ and $q_2$ be the circulating shear flows in cell 1 ($200\times300$ mm) and cell 2 ($300\times300$ mm). The common wall AD carries $q_2-q_1$.
**Areas:** $A_1=200\times300=60\,000$ mm², $A_2=300\times300=90\,000$ mm².
**Equal twist in both cells** ($\theta=\dfrac{1}{2AG}\oint\dfrac{q}{t}ds$ is the same, with the same $t$ and $G$):
$$\frac{q_1(1000)-q_2(300)}{A_1}=\frac{q_2(1200)-q_1(300)}{A_2}$$
where 1000 mm and 1200 mm are the perimeters of the two cells, and 300 mm is the length of the common wall.
$$\frac{1000q_1-300q_2}{60\,000}=\frac{1200q_2-300q_1}{90\,000}\;\Rightarrow\;q_2=\frac{12}{11}q_1 .$$
**Torque:** $T=2(A_1q_1+A_2q_2)$, with $T=12\times10^6$ N mm:
$$12\times10^6=2q_1\left(60\,000+90\,000\times\frac{12}{11}\right)=2q_1(158\,182)\;\Rightarrow\;q_1=37.93\text{ N/mm},\ q_2=41.38\text{ N/mm}.$$
**Shear stress in the common wall** (thickness 1.5 mm):
$$\tau=\frac{q_2-q_1}{t}=\frac{3.45}{1.5}=\mathbf{2.3\ MPa}.$$